2-Group Representations for Spin Foams

Just as 3d state sum models, including 3d quantum gravity, can be built using categories of group representations, ‘2‐categories of 2‐group representations’ may provide interesting state sum models for 4d quantum topology, if not quantum gravity. Here we focus on the ‘Euclidean 2‐group’, built from the rotation group SO (4) and its action on the translation group R4 of Euclidean space. We explain its infinite‐dimensional unitary representations, and construct a model based on the resulting representation 2‐category. This model, with clear geometric content and explicit ‘metric data’ on triangulation edges, shows up naturally in an attempt to write the amplitudes of ordinary quantum field theory in a background independent way.